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Theorem grothpwex 8465
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 8461. Note that ax-pow 4204 is not used by the proof. Use axpweq 4203 to obtain ax-pow 4204. (Contributed by Gérard Lang, 22-Jun-2009.)
Assertion
Ref Expression
grothpwex  |-  ~P x  e.  _V

Proof of Theorem grothpwex
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 8462 . 2  |-  E. y
( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
( z  ~~  y  \/  z  e.  y
) )
2 simpl 443 . . . . . . . 8  |-  ( ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  ->  ~P z  C_  y )
32ralimi 2631 . . . . . . 7  |-  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  ->  A. z  e.  y  ~P z  C_  y
)
4 pweq 3641 . . . . . . . . 9  |-  ( z  =  x  ->  ~P z  =  ~P x
)
54sseq1d 3218 . . . . . . . 8  |-  ( z  =  x  ->  ( ~P z  C_  y  <->  ~P x  C_  y ) )
65rspccv 2894 . . . . . . 7  |-  ( A. z  e.  y  ~P z  C_  y  ->  (
x  e.  y  ->  ~P x  C_  y ) )
73, 6syl 15 . . . . . 6  |-  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  ->  ( x  e.  y  ->  ~P x  C_  y ) )
87anim2i 552 . . . . 5  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
) )  ->  (
x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y
) ) )
983adant3 975 . . . 4  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  /\  A. z  e.  ~P  y ( z 
~~  y  \/  z  e.  y ) )  -> 
( x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y ) ) )
10 pm3.35 570 . . . 4  |-  ( ( x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y
) )  ->  ~P x  C_  y )
11 vex 2804 . . . . 5  |-  y  e. 
_V
1211ssex 4174 . . . 4  |-  ( ~P x  C_  y  ->  ~P x  e.  _V )
139, 10, 123syl 18 . . 3  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  /\  A. z  e.  ~P  y ( z 
~~  y  \/  z  e.  y ) )  ->  ~P x  e.  _V )
1413exlimiv 1624 . 2  |-  ( E. y ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
( z  ~~  y  \/  z  e.  y
) )  ->  ~P x  e.  _V )
151, 14ax-mp 8 1  |-  ~P x  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039    ~~ cen 6876
This theorem is referenced by:  isrnsigaOLD  23488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-groth 8461
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640
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